Question: The interquartile range (IQR) is defined as the difference between the third and the first quartiles. (a) Calculate the IQR of the N(????, ????2) distribution.

The interquartile range (IQR) is defined as the difference between the third and the first quartiles.

(a) Calculate the IQR of the N(????, ????2) distribution.

(b) Consider the sample interquartile range IQR(x) = IQR(x1, x2, ..., xn) = x(⌊3n∕4⌋) − x(⌊n∕4⌋)

as a measure of dispersion. It is known that sample quantiles tend to the respective distribution quantiles if these are unique. Based on this fact, determine the constant c such that the normalized interquartile range IQRN(x) =

IQR (x)∕c is a consistent estimate of ???? when the data has a N(????, ????2)

distribution.

(c) Can you think of a reason why you would prefer MADN(x) to IQRN(x) as a robust estimate of dispersion?

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